To solve the equation \( \sqrt{3x^2 - 10} + 7 = 4 \), we start by isolating the square root term.
Subtract 7 from both sides:
\[ \sqrt{3x^2 - 10} = 4 - 7 \]
This simplifies to:
\[ \sqrt{3x^2 - 10} = -3 \]
Since the square root of a real number cannot be negative, there are no real solutions to this equation.
Thus, the equation \( \sqrt{3x^2 - 10} + 7 = 4 \) has no solutions.